\(\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx\) [19]
Optimal result
Integrand size = 25, antiderivative size = 68 \[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\frac {2 (A-C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 C \tan (c+d x)}{d \sqrt {b \sec (c+d x)}}
\]
[Out]
2*(A-C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/
2)/(b*sec(d*x+c))^(1/2)+2*C*tan(d*x+c)/d/(b*sec(d*x+c))^(1/2)
Rubi [A] (verified)
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of
steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4131, 3856, 2719}
\[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\frac {2 (A-C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 C \tan (c+d x)}{d \sqrt {b \sec (c+d x)}}
\]
[In]
Int[(A + C*Sec[c + d*x]^2)/Sqrt[b*Sec[c + d*x]],x]
[Out]
(2*(A - C)*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*C*Tan[c + d*x])/(d*Sqrt
[b*Sec[c + d*x]])
Rule 2719
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]
Rule 3856
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Rule 4131
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot
[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x
] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Rubi steps \begin{align*}
\text {integral}& = \frac {2 C \tan (c+d x)}{d \sqrt {b \sec (c+d x)}}+(A-C) \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx \\ & = \frac {2 C \tan (c+d x)}{d \sqrt {b \sec (c+d x)}}+\frac {(A-C) \int \sqrt {\cos (c+d x)} \, dx}{\sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \\ & = \frac {2 (A-C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 C \tan (c+d x)}{d \sqrt {b \sec (c+d x)}} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.79 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.85
\[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=-\frac {2 i \left (-3 \left (A+A e^{2 i (c+d x)}-2 C e^{2 i (c+d x)}\right )+2 (A-C) e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )}{3 d \left (1+e^{2 i (c+d x)}\right ) \sqrt {b \sec (c+d x)}}
\]
[In]
Integrate[(A + C*Sec[c + d*x]^2)/Sqrt[b*Sec[c + d*x]],x]
[Out]
(((-2*I)/3)*(-3*(A + A*E^((2*I)*(c + d*x)) - 2*C*E^((2*I)*(c + d*x))) + 2*(A - C)*E^((2*I)*(c + d*x))*Sqrt[1 +
E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(d*(1 + E^((2*I)*(c + d*x)))*Sq
rt[b*Sec[c + d*x]])
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 3.78 (sec) , antiderivative size = 763, normalized size of antiderivative = 11.22
| | |
| method | result | size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(763\) |
| parts |
\(\text {Expression too large to display}\) |
\(778\) |
| | |
|
|
|
|
[In]
int((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
[Out]
2/d/(cos(d*x+c)+1)/(b*sec(d*x+c))^(1/2)*(I*A*EllipticF(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*(
cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-I*A*EllipticE(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2
)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-I*C*EllipticF(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^(
1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+I*C*EllipticE(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1)
)^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+2*I*A*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*EllipticF(I*(cot(d*x+c)-csc(d*x+c)),I)-2*I*A*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2
)*EllipticE(I*(cot(d*x+c)-csc(d*x+c)),I)-2*I*C*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Elli
pticF(I*(cot(d*x+c)-csc(d*x+c)),I)+2*I*C*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(
I*(cot(d*x+c)-csc(d*x+c)),I)+I*A*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(cot(d
*x+c)-csc(d*x+c)),I)*sec(d*x+c)-I*A*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(I*(co
t(d*x+c)-csc(d*x+c)),I)*sec(d*x+c)-I*C*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*
(cot(d*x+c)-csc(d*x+c)),I)*sec(d*x+c)+I*C*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE
(I*(cot(d*x+c)-csc(d*x+c)),I)*sec(d*x+c)+A*sin(d*x+c)+C*tan(d*x+c))
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.46
\[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\frac {\sqrt {2} {\left (i \, A - i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (-i \, A + i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, C \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{b d}
\]
[In]
integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
(sqrt(2)*(I*A - I*C)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)))
+ sqrt(2)*(-I*A + I*C)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c
))) + 2*C*sqrt(b/cos(d*x + c))*sin(d*x + c))/(b*d)
Sympy [F]
\[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\sqrt {b \sec {\left (c + d x \right )}}}\, dx
\]
[In]
integrate((A+C*sec(d*x+c)**2)/(b*sec(d*x+c))**(1/2),x)
[Out]
Integral((A + C*sec(c + d*x)**2)/sqrt(b*sec(c + d*x)), x)
Maxima [F]
\[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\sqrt {b \sec \left (d x + c\right )}} \,d x }
\]
[In]
integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((C*sec(d*x + c)^2 + A)/sqrt(b*sec(d*x + c)), x)
Giac [F]
\[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\sqrt {b \sec \left (d x + c\right )}} \,d x }
\]
[In]
integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate((C*sec(d*x + c)^2 + A)/sqrt(b*sec(d*x + c)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {A+C \sec ^2(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}} \,d x
\]
[In]
int((A + C/cos(c + d*x)^2)/(b/cos(c + d*x))^(1/2),x)
[Out]
int((A + C/cos(c + d*x)^2)/(b/cos(c + d*x))^(1/2), x)